We derive finite-size scaling formulae for four-dimensional Higgs-Yukawa models near the Gaussian fixed point. These formulae will play an essential role in future, detailed investigation of such models. In particular, they can be used to determine the nature of the observed phase transitions, and confirm or rule out the possibility of having non-trivial fixed points in the Higgs-Yukawa models. Our scaling formula for Binders cumulant is tested against lattice simulations carried out at weak couplings, and good agreement is found. As a separate project, we also present preliminary results from our study of a chirally-invariant Higgs-Yukawa model including a dimension-six operator at finite temperature. Our work provides first indications of first-order temperature-induced phase transitions near the infinite cutoff limit in this model.
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